Theorem sbalex 2280

Description: Equivalence of two ways to express proper substitution of a setvar for another setvar disjoint from it in a formula. This proof of their equivalence does not use df-sb 2094.

That both sides of the biconditional express proper substitution is proved by sb5 2313 and sb6 2121. The implication "to the left" is equs4v 2023 and does not require ax-10 2178 nor ax-12 2215. It also holds without disjoint variable condition if we allow more axioms (see equs4 2450). Theorem 6.2 of [Quine] p. 40. Theorem equs5 2494 replaces the disjoint variable condition with a distinctor antecedent. Theorem equs45f 2493 replaces the disjoint variable condition on 𝑥, 𝑡 with the nonfreeness hypothesis of 𝑡 in 𝜑. (Contributed by NM, 14-Apr-2008.) Revised to use equsexv 2306 in place of equsex 2452 in order to remove dependency on ax-13 2406. (Revised by BJ, 20-Dec-2020.) Revise to remove dependency on df-sb 2094. (Revised by BJ, 21-Sep-2024.) (Proof shortened by SN, 14-Aug-2025.)

Assertion

Ref	Expression
sbalex	⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Distinct variable group:   𝑥,𝑡

Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sbalex

Step	Hyp	Ref	Expression
1		nfe1 2187	. . 3 ⊢ Ⅎ𝑥∃𝑥(𝑥 = 𝑡 ∧ 𝜑)
2		ax12ev2 2218	. . 3 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → (𝑥 = 𝑡 → 𝜑))
3	1, 2	alrimi 2251	. 2 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
4		equs4v 2023	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∃𝑥(𝑥 = 𝑡 ∧ 𝜑))
5	3, 4	impbii 212	1 ⊢ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1561  ∃wex 1802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-10 2178  ax-12 2215

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1803  df-nf 1807

This theorem is referenced by:  sb5  2313  dfsb7  2316  alexeqg  3613  dfdif3OLD  4075  regsfromsetind  36912  pm13.196a  44988